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June  2012, 5(3): 641-656. doi: 10.3934/dcdss.2012.5.641

An explicit stable numerical scheme for the $1D$ transport equation

1. 

Commissariat à l’Énergie Atomique (CEA), DEN/DANS/DM2S/SFME/LETR, 91191 Gif-sur-Yvette, France

Received  August 2010 Revised  October 2010 Published  October 2011

We derive in this paper a numerical scheme in order to calculate solutions of $1D$ transport equations. This $2nd$-order scheme is based on the method of characteristics and consists of two steps: the first step is about the approximation of the foot of the characteristic curve whereas the second one deals with the computation of the solution at this point. The main idea in our scheme is to combine two $2nd$-order interpolation schemes so as to preserve the maximum principle. The resulting method is designed for classical solutions and is unconditionally stable.
Citation: Yohan Penel. An explicit stable numerical scheme for the $1D$ transport equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 641-656. doi: 10.3934/dcdss.2012.5.641
References:
[1]

C. Bardos, M. Bercovier and O. Pironneau, The vortex method with finite elements, Math. Comp., 36 (1981), 119-136. doi: 10.1090/S0025-5718-1981-0595046-3.  Google Scholar

[2]

F. Boyer and P. Fabrie, "Éléments d'Analyse pour l'Étude de Quelques Modèles d'Écoulements de Fluides Visqueux Incompressibles,'' Mathématiques & Applications (Berlin), 52, Springer-Verlag, Berlin, 2006.  Google Scholar

[3]

J. Burgers, "A Mathematical Model Illustrating the Theory of Turbulence," edited by Richard von Mises and Theodore von Kármán, Adv. Appl. Mech., Academic Press, Inc., New York, (1948), 171-199. doi: 10.1016/S0065-2156(08)70100-5.  Google Scholar

[4]

S. Dellacherie, On a diphasic low Mach number system, M2AN Math. Model. Numer. Anal., 39 (2005), 487-514. doi: 10.1051/m2an:2005020.  Google Scholar

[5]

B. Després and F. Lagoutière, Contact discontinuity capturing schemes for linear advection and compressible gas dynamics, J. Sci. Comput., 16 (2001), 479-524. doi: 10.1023/A:1013298408777.  Google Scholar

[6]

J. Douglas Jr. and T. Russell, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal., 19 (1982), 871-885.  Google Scholar

[7]

J. Douglas Jr., C.-S. Huang and F. Pereira, The modified method of characteristics with adjusted advection, Numer. Math., 83 (1999), 353-369. doi: 10.1007/s002110050453.  Google Scholar

[8]

G. Fourestey, "Simulation Numérique et Contrôle Optimal d'Interactions Fluide Incompressible/Structure par une Méthode de Lagrange-Galerkin d'Ordre 2,'' Ph.D Thesis, École Nationale des Ponts et Chaussées, 2002. Available at: http://hal.archives-ouvertes.fr/tel-00005675. Google Scholar

[9]

E. Godlewski and P.-A. Raviart, "Numerical Approximation of Hyperbolic Systems of Conservation Laws,'' Applied Mathematical Sciences, 118, Springer-Verlag, New York, 1996.  Google Scholar

[10]

F. Holly and A. Preissmann, Accurate calculation of transport in two dimensions, J. Hydr. Div., 103 (1977), 1259-1277. Google Scholar

[11]

R. LeVeque, "Numerical Methods for Conservation Laws," Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992.  Google Scholar

[12]

J. Marsden and A. Chorin, "A Mathematical Introduction to Fluid Mechanics," Springer-Verlag, New-York-Heidelberg, 1979.  Google Scholar

[13]

S. Osher and J. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2.  Google Scholar

[14]

Y. Penel, "Étude Théorique et Numérique de la Déformation d'une Interface Séparant deux Fluides Non-Miscibles à Bas Nombre de Mach,", Ph.D Thesis, ().   Google Scholar

[15]

Y. Penel, S. Dellacherie and O. Lafitte, Global solutions to the 1D Abstract Bubble Vibration model,, Submitted., ().   Google Scholar

[16]

O. Pironneau, On the transport-diffusion algorithm and its applications to the Navier-Stokes equations,, Numer. Math., 38 (): 309.  doi: 10.1007/BF01396435.  Google Scholar

show all references

References:
[1]

C. Bardos, M. Bercovier and O. Pironneau, The vortex method with finite elements, Math. Comp., 36 (1981), 119-136. doi: 10.1090/S0025-5718-1981-0595046-3.  Google Scholar

[2]

F. Boyer and P. Fabrie, "Éléments d'Analyse pour l'Étude de Quelques Modèles d'Écoulements de Fluides Visqueux Incompressibles,'' Mathématiques & Applications (Berlin), 52, Springer-Verlag, Berlin, 2006.  Google Scholar

[3]

J. Burgers, "A Mathematical Model Illustrating the Theory of Turbulence," edited by Richard von Mises and Theodore von Kármán, Adv. Appl. Mech., Academic Press, Inc., New York, (1948), 171-199. doi: 10.1016/S0065-2156(08)70100-5.  Google Scholar

[4]

S. Dellacherie, On a diphasic low Mach number system, M2AN Math. Model. Numer. Anal., 39 (2005), 487-514. doi: 10.1051/m2an:2005020.  Google Scholar

[5]

B. Després and F. Lagoutière, Contact discontinuity capturing schemes for linear advection and compressible gas dynamics, J. Sci. Comput., 16 (2001), 479-524. doi: 10.1023/A:1013298408777.  Google Scholar

[6]

J. Douglas Jr. and T. Russell, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal., 19 (1982), 871-885.  Google Scholar

[7]

J. Douglas Jr., C.-S. Huang and F. Pereira, The modified method of characteristics with adjusted advection, Numer. Math., 83 (1999), 353-369. doi: 10.1007/s002110050453.  Google Scholar

[8]

G. Fourestey, "Simulation Numérique et Contrôle Optimal d'Interactions Fluide Incompressible/Structure par une Méthode de Lagrange-Galerkin d'Ordre 2,'' Ph.D Thesis, École Nationale des Ponts et Chaussées, 2002. Available at: http://hal.archives-ouvertes.fr/tel-00005675. Google Scholar

[9]

E. Godlewski and P.-A. Raviart, "Numerical Approximation of Hyperbolic Systems of Conservation Laws,'' Applied Mathematical Sciences, 118, Springer-Verlag, New York, 1996.  Google Scholar

[10]

F. Holly and A. Preissmann, Accurate calculation of transport in two dimensions, J. Hydr. Div., 103 (1977), 1259-1277. Google Scholar

[11]

R. LeVeque, "Numerical Methods for Conservation Laws," Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992.  Google Scholar

[12]

J. Marsden and A. Chorin, "A Mathematical Introduction to Fluid Mechanics," Springer-Verlag, New-York-Heidelberg, 1979.  Google Scholar

[13]

S. Osher and J. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2.  Google Scholar

[14]

Y. Penel, "Étude Théorique et Numérique de la Déformation d'une Interface Séparant deux Fluides Non-Miscibles à Bas Nombre de Mach,", Ph.D Thesis, ().   Google Scholar

[15]

Y. Penel, S. Dellacherie and O. Lafitte, Global solutions to the 1D Abstract Bubble Vibration model,, Submitted., ().   Google Scholar

[16]

O. Pironneau, On the transport-diffusion algorithm and its applications to the Navier-Stokes equations,, Numer. Math., 38 (): 309.  doi: 10.1007/BF01396435.  Google Scholar

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